"WM" <mueckenh@rz.fh-augsburg.de> wrote in message news:b6fe23bb-f753-4cfc-a9d6-79711f3a1e5d@r40g2000yqn.googlegroups.com...> On 9 Dez., 08:35, Virgil <Vir...@home.esc> wrote:>> In article >> <c8idnc6JG6hv34LWnZ2dnUVZ_q-dn...@giganews.com>,>>>> However, in the discussion between Dik and WM, Dik gave >> SPECIFIC>> definition of what HE meant by the limit of a seqeunce of >> sets which>> differed from that in your citation.>> Dik:> Given a sequence of sets S_n then:> lim sup{n -> oo} S_n contains those elements that occur > in> infinitely> many S_n

I don't think this is a good definition for the limit of a sequence of sets S_n. For instance, consider the alternating sets:

To me, I don't see the S_n converging to anything, certainly not the set of negative odd numbers union the set of even numbers. Like Thompson's lamp alternating between on and off it reaches no limit in any intuitive sense of what a limit is.

> lim inf{n -> oo} S_n contains those elements that occur > in all S_n> from> a certain S_n (which can be different > for each> element).> lim{n -> oo} S_n exists whenever lim sup and lim inf are > equal.>>>> > Define infimum and supremum as follows:>>>> > liminf X_n=\/(n=0, oo)[/\(m=n, oo) X_m]>> > (n-->oo)>>>> > limsup X_n=/\(n=0, oo)[\/(m=n, oo) X_m]>> > (n-->oo)>>>> > If these two are the same then the limit exists and is >> > both>> > of them.>>>> The issue is not whether the naturals are such a limit >> but whether for>> every so defined limit the cardinality of the limit >> equals the limit>> cardinality of their cardinalities, which is different >> sort of limit>> If the limit set emerges continuously (i.e. growing > one-by-one) from> the sets of the sequence, then the cardinality will be > bound to this> process and will also continuously emerge from the > cardinalities of> the sets of the sequence.>> If there is a gap in one sequence and no gap in the other, > then this> indicates that at least one of the limits does not exist, > probably> both do not exist.>> This has been substanciated by the following > thought-experiment:> If N is generated as a limit, then it is generated by > adding number> after number.> When using an intermediate reservoir, as shown in my > lesson> http://www.hs-augsburg.de/~mueckenh/GU/GU12.PPT#394,22,Folie> 22> it becomes clear that N cannot be generated by adding > number after> number.

Why not? Say we have an infinitely large sheet of paper and we print each natural number, n, on the paper at time t=1-1/(n+1). Certainly at time t=1 we have all the natruals printed on the page.